Several basic statistics questions

Hi can you please help me with some basic questions? I'm self-teaching and I want to make sure I'm not messing up!
1. When you have the sampling distribution of the mean of any distribution, by applying the central limit theory you can calculate the standard error by sigma (or estimated sigma, if you don't know sigma) by the root of the # of your observations. When you keep your sample size the same and increase the number of trials, you approach a normal distribution with a smaller and smaller SD of the mean. If your sample size is 1 you don't get a nice bell shape for the mean sample size though if your population is not bell-shaped. Like when you have a sample of 1 you're just going to get closer and closer to the actual distribution. Is what I wrote right?
2. If your population is not bell-shaped (let's say very positively skewed), what minimum sample size do you need to get a bell-shaped distribution of the sample mean? Can it be as small as 2 as long as you have A LOT of simulations (a million, 5 million, 10 million, maybe more)?

Re: Several basic statistics questions

You don't pick a sample size and repeatedly find the mean with many sets of data with that sample size. You put all of your samples into one data set and the size of that is the sample size that affects the accuracy of the sample mean. Does this clear things up?

Re: Several basic statistics questions

Re: Several basic statistics questions

Yes it is correct from a theoretical standpoint. If you collected 50 samples and found the sample mean, then 50 more samples and found the new sample mean, and repeated that again and again; if you plotted all your sample means on a histogram then it would look like a bell curve.

As for your second question, if you are not sampling from a bell-shaped population then usually it is recommended to have 30 or 50 samples before using the normal distribution approximation. For some distributions there are better rules of thumb to use for the sample size.
If you were sampling from a binomial distribution where the value can only be 1 or 0 then if you just took 2 samples the sample mean would be either 0 or 0.5 or 1. So if you repeatedly took 2 samples 1 million times then the histogram would just be three points and not like a bell curve at all.